On the convergence of numerical schemes for the Boltzmann equation

2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved 10.1016/S0294-1449(02)00029-X/FLA

ON THE CONVERGENCE OF NUMERICAL SCHEMES FOR THE BOLTZMANN EQUATION

T. HORSIN^a,∗, S. MISCHLER^a, A. VASSEUR^b

aLaboratoire de mathématiques appliquées, UMR7641, Université de Versailles – Saint Quentin, 45, avenue des Etats-Unis, 78035 Versailles cedex, France

bLaboratoire J.A. Dieudonné, Université de Nice-Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, France

Received 13 March 2002

ABSTRACT. – We consider a time and spatial explicit discretisation scheme for the Boltzman- nequation. We prove some Maxwellian bounds on the resulting approximated solution anddeduce its convergence using a new time-discrete averaging lemma.

2003 Éditions scientifiques et médicales Elsevier SAS MSC: 35A35; 65L20; 76PO5

RÉSUMÉ. – Nous considérons une discrétisation explicite en temps et espace de l’équation de Boltzmann. Nous exhibons des bornes au moyen de Maxwelliennes et en déduisons la convergence en utilisant un nouveau lemme de moyennisation discret en temps.

2003 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

This article is devoted to the proof of the convergence of a time and spatial explicit discretisation scheme for the Boltzmann equation. The Boltzmann equation provides a time evolution of a gas described by the distribution of particlesf (t, x, v)0 which at timet0 and at positionx∈R^d move with velocityv∈R^d. The Boltzmann equation reads

∂f

∂t +v· ∇xf =Q(f, f ) in(0,∞)×R^d×R^d, (1.1) f (0, x, v)=fin(x, v) onR^d×R^d, (1.2) where Q(f, f ) is the quadratic Boltzmann collision operator describing the collision interactions between particles (binary elastic shock). We refer to [3] for a detailed

∗Corresponding author.

E-mail addresses: horsin@math.uvsq.fr (T. Horsin), mischler@math.uvsq.fr (S. Mischler), vasseur@math.unice.fr (A. Vasseur).

presentation of the equation and to [27] and the references therein for recent results concerning its analysis. Let us just summarize now the fundamental properties of the collision kernel that we shall use in the sequel. First, the collision kernel splits into two parts

Q(f, f )=Q⁺(f, f )−Q⁻(f, f )

where the gain term Q⁺and the loss term Q⁻are positive operators. Next, it vanishes on Maxwellian functions, namely

Q⁺(M, M)=Q⁻(M, M) ifM(v)=expa|v|²+b·v+c (1.3) witha, c∈R,a <0,b∈R^d. Last, the loss term writes

Q⁻(f, f )=f L(f ), L(f )=A∗vf, (1.4) and we assume here that the so-called total cross-sectionAsatisfies

0A(z)K0|z|^γ, K0>0, γ ∈(−d,1]. (1.5) A particular case for which the above condition holds is the cross-section associated to an inverse potential (except the Coulomb potential) with the angular cut-off condition of Grad and the cross-section associated to hard sphere collisions. Before describing the scheme investigated here, we recall what is known about several partial discretisations of (1.1). A first step of the discretisation (usually used in numerical simulation) is to split the transport part

∂f

∂t +v· ∇xf =0 (1.6)

and the collision part

∂f

∂t =Q(f, f ), (1.7)

and to solve each equation one after another in small intervals (kt, (k +1)t) for any k∈N. This splitting algorithm has been proved to converge in [4]: constructing a approximate solutionft, one may prove that(ft)converges (up to the extraction of a subsequence) to a solution of the Boltzmann equation (1.1) whent→0. A crucial step is the velocity discretisation, which means to approximate (1.1) by a family of equations

∂fj

∂t +vj· ∇xfj =Qj

(fj)j, (fj)j

in(0,∞)×R^d, (1.8)

where fj =fj(t, x)0 represents the density of particles with velocity vj (or with velocity in a neighborhood ofvj), (vj)is a family of given velocities and the operator (Qj)j is an approximation of Q(f, f ) (with the help of quadrature formula). Such a scheme (construction of a good approximation operatorQj) have been proposed by [10, 15,21] and their convergence have been proved in [17,19,21,20,16].

Another step is to perform a time Euler explicit discretisation of (1.7):

f^k+1−f^k t

=QRv

f^k, f^k, f⁰=fin. (1.9)

HereQRv denote a velocity truncation ofQ which guarantees the positivity off^k and can be relaxed in the limit t →0. Convergence of the Euler scheme has been proved in [18]. See also [9] for other time discretisations.

The scheme we consider here consists in an explicit time and space discretisation of the splitting algorithm of (1.1). Full discretisations including velocity discretisation is postponed to future works. We successively perform (and iterate):

(1) solve explicitly the transport equation (1.6), (2) project on space mesh,

(3) perform the time explicit Euler scheme (1.9).

In order to be more precise, let us introduce a partition ofR^din cells:

R^d=

a∈Z^d

a, a= d

i=1

aix,n, (ai+1)x,n

, (1.10)

for somex,n>0; and let us define the projection operator on the meshes(a)_a∈Z^d: Pnφ=

a

P^aφ withP^aφ(x):= 1 (x,n)^d

_a

φ(y) dy1_a(x). (1.11)

Let also defineQRv,n a velocity truncated Boltzmann operator such that its total cross- sectionARv,n satisfies

ARv,n(z)A(z)1_|z|Rv,n. Starting from the initial datum

f_n⁰=(Pnfin)1B_Rv,n(v)1B_Rx,n/4(x) (1.12)

we define

(f_n^k+1/3)^#(x, v)=f_n^k(x, v), f_n^k+2/3=Pnf_n^k+1/3,

f_n^k+1−f_n^k+2/3 t,n

=QRv,n

f_n^k+2/3, f_n^k+2/3

where we use the notationφ^{k #}(x, v):=φ(x+kt,nv, v)fork∈Z(for a givent,n>0).

In other words, we define f_n^k+1=Pn

f_n^k−#+t,nQRv,n

Pn

f_n^k−#, Pn

f_n^k−#. (1.13)

We finally define the approximate solutionfnby fn(t, x, v)=

k

f_n^kx−v(t−kt,n), v1t∈[kt,n,(k+1)t,n)1t∈[0,Tn] (1.14) for a given choice oft,n,x,n,Rx,n,Rv,n,Tn>0.

This paper is devoted to the proof of the following result.

THEOREM 1.1. – Let consider an initial datumf_insuch that

0finM⁰=C⁰ξ , ξ(x, v)=exp−α|x|²−β|v|², (1.15) with

(i) local case: γ ∈(−d,0]

or

(i) global case: γ ∈(−d+1,0]andC⁰small enough (depending onα, β) orγ ∈(0,1]andαlarge enough (depending onC⁰, β).

There exists T^∗=T^∗(M⁰) >0, and we may choose T^∗= +∞ in case (ii), and there exists a sequence of the discretisation parameters (t,n), (x,n), (Rx,n), (Rv,n), (Tn) satisfying

t,n, x,n→0, Rx,n, Rv,n→ +∞, TnT^∗, (1.16) such that the sequence(fn)defined by (1.14) satisfies

sup

n [sup0,T^∗]

fnξ⁻¹_L∞<∞, (1.17)

and, up to the extraction of a subsequence,(fn)converges weakly to a solutionf of the Boltzmann equation (1.1).

Remark 1.2. – The same result holds for different versions of time and space discretisations such that replacing (1.13) by

f^k+1#=Pn

f^k+t,nQRv,n

Pn

f^k, Pn

f^k.

Let us briefly explain the strategy of the proof. First, remark that though the convergence proof for the splitting algorithm and for the velocity discretisation scheme can be performed in the general framework of DiPerna–Lions renormalized solutions (and thus for general initial data) such a framework seems difficult to use in the present situation at least for two reasons. On one hand, for an explicit scheme we loose the entropy-dissipation entropy bound which is a fundamental information in the weak stability result for renormalized solutions. On the other hand, even for a modified implicit scheme (for which entropy-dissipation entropy bound is available) time (and position) discretisation seems to be inadapted to the renormalization technic. We then choose the (less general) framework of distributional solution bounded above by a Maxwellian function introduced by Illner and Shinbrot.

The first step is thus to build for anyn∈Na sequence of Maxwellians(M_n^k)k which are subsolution of the discrete scheme (1.13) in the following sense

M_n^k+1PnM_n^k−#+t,nQ⁺_Rv,nPnM_n^k−#, PnM_n^k−# (1.18) in the case of soft potential (γ 0) and

M_n^kπ_n^kM⁰+

k−1

j=0

t,nπ_n^k−1−jQ⁺_Rv,nπnM_n^j, πnM_n^j, (1.19)

in the case of hard potential (γ >0), where we use the notationπnφ=Pn(φ^−#). These sub-solutions(M_n^k)can be constructed locally or globally in time (depending on the size of the initial datum and ofγ). We then easily verify that they are indeed subsolutions: if 0f_n⁰M_n⁰then 0f_n^kM_n^k ∀k, nand that provides the strong bounds (1.17).

A second step is to write the kinetic equation satisfied byfn, namely

∂fn

∂t +v· ∇xfn=

k

δkt,n(t)

(k+1)t,n

kt,n

Pngn−gn

t,n

+Q_Rv,n(Pngn, Pngn)

dτ (1.20)

with

gn(t,·,·)=fn

(tt,n)⁻,·,·, tt,n=E(t/t,n)t,n, (1.21) where E denotes the truncation function, and to pass to the limit in (1.20) when n→ ∞. In order to do it, the main difficulty is to prove that the velocity averages of gn converge strongly. Of course, the so-called “compactness lemma on velocity averaging” of solutions of continuous transport equation has been introduced by [12, 11,1] at the middle of the 80’s and has been extensively developped by [5,6,8,22,2].

Discrete versions in velocity have been proved in [17] and time discrete version for the splitting algorithm have been introduced in [4]. See also [2] for an alternative and simpler proof. We need here such a discrete version of averaging lemmas (which means for velocity averaging ofgn instead of velocity averaging of fn) extended to this time and position discrete context. Gathering the “ultimate” version of averaging lemma due to [22], the previous “time” discrete version of averaging lemma by [4] and [2] and the scale techniques developed by Vasseur in [25,26], we prove the following result.

THEOREM 1.3. – Consider a sequence t,n → 0 and a sequence fn uniformly bounded inL^∞(R⁺×R^2d)which satisfies

∂

∂tfn+v· ∇xfn=

i∈Z

δi_t,n(t)

⁽ⁱ+1)_t,n i_t,n

Hn(τ, x, v) dτ

=Jn. (1.22)

We assume that

fn0 f weakly inL^∞(R⁺×R^2d)∗ (1.23)

Jnis relatively compact inW^−1,p(R⁺×R^2d)for somep >1, (1.24) there exists a sequence εn→0 withεn/t,n→ +∞such that: (1.25)

εn2Hn

L²

n−→→+∞0.

Then, for anyψ∈D(R^d),

R^d

gn(t, x, v)ψ(v) dv →

R^d

f (t, x, v)ψ(v) dv (1.26)

strongly inL^p_loc((0, T )×R^2d)∀p∈ [1,∞).

It remains to verify that Theorem 1.3 may be used for the sequence(fn)built in the statement of Theorem 1.1, and then it is classical to pass to the limit n→ ∞ in the formulation (1.20) and obtain Theorem 1.1. For the sake of completeness we present in the appendix a different version of Theorem 1.3 where hypothesis (1.24) is slightly generalized.

The outline of the paper is the following. In Section 2 we prove Theorem 1.3. In Section 3 we built the subsolution(Mn)for the discrete scheme (1.13). In Section 4 we then prove Theorem 1.1.

2. Proof of Theorem 1.3

Let us begin giving the idea of the proof. We first use the classical compactness averaging lemma to prove compactness for the continuous functions with respect to time. Indeed we are able to show that {ψ(v)fn(·,·, v) dv} is relatively compact in L²_loc(R^d+1). Let us recall this result due to Perthame and Souganidis [22] in our framework:

THEOREM 2.1. – Letfn be a sequence of functions bounded inL^q(R^1+2d)for some 1< q <+∞and{Jn}be relatively compact inW^−1,p(R^1+2d)verifying

∂tfn+v.∇xfn=Jn. Then, for every functionψ∈D(R^d), the average

ρ_ψⁿ(t, x):=

R^d

ψ(v)fn(t, x, v) dv

is relatively compact inL^q(R^1+d).

On the other hand, property (1.25) allows us to show a result of the kind (1.26) “at a local scale” thanks to the following Theorem due to Desvillettes and Mischler [4].

THEOREM 2.2. – Consider a sequence ¯n →0 and a sequence of functions f¯n

bounded inL²_loc([0, T] ×R^2d)which verify

∂sf¯n+v· ∇yf¯n=

i∈Z

δ_i¯n(s)

(i+1)¯_n i¯n

H¯n(τ, y , v) dτ

(2.1)

withH¯nbounded inL²([0, T] ×R^2d). Then, for everyψ∈D(R^d), the average

¯

η_ψⁿ(s, y)=

R^d

ψ(v)g¯n(s, y , v)dv

is relatively compact inL²_loc([0, T] ×R^d), where

¯

gn(s, y , v)= ¯fn(s_¯n, y , v),

withs_¯n=E(s/¯n)¯n.

More precisely, if we denote¯n=t,n/εnwitht,nεn1, and f¯n(t, x, s, y , v)=fn(tt,n+sεn, x+yεn, v),

¯

gn(t, x, s, y , v)=gn(t_t,n+sεn, x+yεn, v)=fn(t_t,n+εns_¯n, x+yεn, v), then for every fixed point(t, x)the functionf¯n(t, x,·,·,·)verifies (2.1). So we conclude that {ψ(v)g¯n(t, x,·,·, v) dv} is relatively compact inL²([0, T] ×R^d) when (t, x)is fixed. The following lemma allows us to compare the results at the global scale (in variables (t, x)) and at the local scale (in variables (s, y)) in order to carry the desired result from the result at the local scale using the compactness result on the continuous function in time at the global scale:

LEMMA 2.3 (From local scale to global scale). – Let ρⁿ, ρ ∈ L^p_loc([0, T] ×R^d) with 1p <+∞, t,n→0 and t,n/εn→0. Then ρⁿ converges strongly to ρ in L^p_loc([0, T] ×R^d)if and only if for everyR >0:

T

0

Bd(0,R)

1

0

Bd(0,1)

ρⁿ(tt,n+εns, x+εny)−ρ(t, x)^pdyds dx dtⁿ−→^→+∞0, (2.2)

whereBd(0, R)is thed-dimensional ball of center 0 and radius R.

This lemma is a slight generalization of a result of [25] (in the caseεn=t,n). For the sake of completeness we give its proof in Appendix A.

Proof of Theorem 2. – We denote:

ηⁿ_ψ(t, x)=

R^d

ψ(v)gn(t, x, v) dv.

We split the proof into several parts.

(i) Compactness at the global scale. For everyj∈Nwe consider a regular function 9j ∈C^∞(R×R^d)defined such that Supp9j ⊂ [0,2j] ×Bd(0,2j )and9j(t, x)=1 if (t, x)∈ [1/j, j] ×Bd(0, j ). From Eq. (1.22) we get:

∂9jfn

∂t +v· ∇x9jfn=9jJn+fn

∂t9j+v· ∇x9j

.

Sincefn∈L^∞and9jis regular the right-hand-side term is compact inW^−1,p(R×R^2d).

Moreover 9jfn ∈L^p(R×R^2d) (since fn ∈ L^∞ and 9j is compactly supported).

Therefore {9j(t, x) ψ(v)fn(t, x, v) dv} is relatively compact in L^p(R^1+2d) thanks to Theorem 2.1. By diagonal extraction, up to a subsequence, there exists ρψ ∈ L^∞([0, T]×R^d)such thatρ_ψⁿ(t, x)=ψ(v)fn(t, x, v) dvconverges toρψ(t, x)inL^p_loc. Because of hypothesis (1.23), ρψ(t, x)=ψ(v)f dv. By uniqueness of the limit the entire sequence is converging. Finally the convergence holds true inL^q_loc([0, T] ×R^2d) for 1q <+∞as well since

sup

n0

ρ_ψⁿ_L∞sup

n0

fnL^∞ψL¹<∞.

In short we have proved

ρ_ψ^l −→ f ψ dv (2.3)

inL^p_loc([0, T] ×R^d, ∀p∈ [1,+∞[.

(ii) From global scale to local scale.

We consider the local functions depending on the local variables(s, y). We introduce the two new ones:

¯

ρ_ψⁿ(t, x, s, y )=

ψ(v)f¯n(t, x, s, y , v) dv,

¯

η_ψⁿ(t, x, s, y )= ψ(v)g¯n(t, x, s, y , v) dv.

From Lemma 2.3 and (2.3), we deduce that for everyR, T >0, 1p <+∞: T

0

Bd(0,R)

1 0

Bd(0,1)

ρ¯_ψⁿ(t, x, s, y )−ρψ(t, x)^pds dy

dx dtⁿ−→^→+∞0.

From hypothesis (1.25), _Rd|εn2Hn|²(·,·, v) dv converges to 0 in L¹(R⁺×R^d). Then Lemma 2.3 withp=1 implies that:

T

0

B_d(0,R) R^d

1

0

B_d(0,1)

H¯n(t, x, s, y , v)²ds dydv

dx dt^n→+∞−→ 0,

where we denote H¯n = εn2Hn(tt,n +εns, x +εny , v). This leads to the following proposition:

PROPOSITION 2.4. – Up to a subsequence (still denoting t,n), there exists ;⊂ [0, T] ×R^d withL([0, T] ×R^d\;)=0 such that for every(t, x)∈;:

1

0

B_d(0,1)

ρ¯_ψⁿ(t, x, s, y )−ρψ(t, x)^pds dyⁿ−→^→+∞0,

R^d

1

0

Bd(0,1)

H¯n(t, x, s, y , v)²ds dydvⁿ−→^→+∞0,

for every 1p <+∞, whereLdenotes the Lebesgue measure.

From now on we fix(t, x)∈;.

(iii) Strong convergence at the local scale.

Since the point(t, x)∈;is fixed, let us skip it in the notation (so we denotef¯n(s, y , v) forf¯n(t, x, s, y , v)). We have:

LEMMA 2.5. – Local functionsf¯nare bounded inL^∞([0, T] ×R^2d)and verify:

∂f¯n

∂s +v.∇yf¯n=

i∈Z

δ_i¯n(s)

(i+1)¯n

i¯n

H¯ε_n(τ, y , v) dτ

. (2.4)

Proof. – We just compute:

∂f¯n

∂s +v.∇yf¯n=εn

i∈Z

δi_t,n(t_t,n+εns)

⁽ⁱ+1)t,n

it,n

Hn(τ, x+εny , v) dτ

=

i∈Z

δi_t,n(εns)

⁽ⁱ+1)t,n+t_t,n i_t,n+t_t,n

εnHn(τ, x+εny , v) dτ

=

i∈Z

δi_t,n(εns)

(i+1)¯_n i¯n

εn2Hn(t_t,n+εnτ, x+εny , v) dτ

=

i∈Z

δ_i¯n(s)

(i+1)¯n

i¯n

H¯εn(τ, y , v) dτ

.

In the second equality we do the change of indicei→i+tt,n/t,n, in the third equality we do the change of variables τ →t_t,n +εnτ and in the last equality we use the definition ofH¯εnand the remark thatεns=it,nif and only ifs=i¯n. ✷

This lemma gives the hypothesis needed to apply Theorem 2.2 (with Proposition 2.4).

Therefore we conclude that:

PROPOSITION 2.6. – The sequence{ ¯η_ψⁿ}is relatively compact inL²_loc([0, T] ×R^d).

(iv) Uniqueness of the limit at the local scale.

PROPOSITION 2.7. – The entire sequence η¯_ψⁿ(s, y) converges to ρψ (which is constant with respect to(s, y)) inL²_loc([0, T] ×R^d)whenngoes to+∞.

Proof. – We have

fn(t, x, v)=fn(tt,n, x+(t−tt,n)v, v)

=gn(t, x+(t−t_t,n)v, v).

Notice that

(t_t,n+εns)_t,n=t_t,n+εns_¯n, Sof¯n(s, y , v)= ¯gn(s, y−εns_¯nv, v)and

¯

ρ_ψⁿ(s, y)− ¯η_ψⁿ(s, y)=

R^d

ψ(v)g¯n(s, y−εns_¯nv, v)− ¯gn(s, y , v)dv.

If we consider a test functionφ∈C_c^∞([0, T] ×R^2d)we find:

φ(s, y)ρ¯_ψⁿ(s, y)− ¯ηⁿ_ψ(s, y)ds dy

= ψ(v)g¯n(s, y , v)φ(s, y)−φ(s, y+εns_¯nv)ds dydv t,n∇φL^∞(Suppφ)ψL¹ ¯fnL^∞

n−→→∞0.

Therefore:

¯

ηⁿ_ψ− ¯ρ_ψⁿ −→^D 0.

Since ρ¯_ψⁿ converges toρψ thanks to Proposition 2.4, the entire sequence η¯ⁿ_ψ converges to ρψ in the sense of distribution and we conclude gathering this information with Proposition 2.6. ✷

(v) Back to the global scale.

We have shown that for every(t, x)∈;:

1

0

Bd(0,1)

η¯ⁿ_ψ(t, x, s, y )−ρψ(t, x)²ds dy^n→+∞−→ 0.

Therefore, sinceL([0, T] ×R^d\;)=0, for every 1p <2:

T

0

Bd(0,R)

1

0

Bd(0,1)

η¯ⁿ_ψ(t, x, s, y )−ρψ(t, x)^pds dydx dtⁿ−→^→+∞0.

Using Lemma 2.3 we conclude thatηⁿ_ψ converges toρψ inL^p_loc([0, T] ×R^d)which ends the proof of Therorem 1.3. ✷

3. Subsolution

In this section we fix some positive realt, x,Rx,Rv,T (without dependence in n) and we prove several estimates on the sequence (f^k)defined by the discrete scheme (1.13). We treat separatly the case of the soft potential and the case of the hard potential.

In all what follows we define

ξ(x, v)=exp−α|x|²−β|v|², ξ^{k #}(x, v)=exp−α|x+vkt|²−β|v|². (3.1) Case 1. Soft potentialγ ∈(−d,0]. We begin with some technical lemmas that we will use in the construction of subsolutions.

LEMMA 3.1. – There existsK1=K1(α, β, γ )such that 0Lξ^{−k #}<(kt)

4 ∀x, v∈R^d, kk^∗:=E T t

, (3.2)

where

<(t):= K1

(1+t)^d+γ. Proof. – We write

ξ^{−k #}=exp−A(kt)x−B(kt)v²−C(kt)|x|² with

C(t)= αβ

αt²+β, B(t)=αt²+β and A(t)= tα αt²+β. By a change of variables, we have

ξ^{−k #}∗v|.|^γ(v)=e^−C|x|2

R^d

exp−Ax−B(v−z)²|z|^γdz

=e^−C|x|2 B^γ+d

R^d

exp−Ax−Bv−w²|w|^γdw

=e^−C|x|2 B^γ+d

e^−|.|2∗ |.|^γ(Ax−Bv).

We conclude that

Lξ^{−k #}_L∞(Rd

v) K0

Cγ

B^γ+d

since|.|^γ ∈L¹(R^d)+L^∞(R^d)and e^−|.|2 ∈L¹(R^d)∩L^∞(R^d). ✷

LEMMA 3.2. – For anyC⁰>0 there existsT , C^∗, τ >0 (depending on C⁰, γ +d, K1) such that the sequence(C^k)k1recursively defined by

C^k+1=(1+tτ )C^k+t

C^k2<(k+1)t

(3.3)

satisfies

0C^kC^∗ for anyk=0,1, . . . , k^∗. (3.4) Moreover, there exists C¯ = ¯C(d+γ , K1)∈(0,1) such that if γ ∈ (−d +1,0] and C⁰∈(0,C), then, for any¯ T >0, setting

τ = K2

(1+T )^γ+d, K2=K2(K1, d, γ ) >0, (3.5) the sequence (C^k)satisfies (3.4) withC^∗=1.

Remark 3.3. – We define the interval(0, T^∗)on which a uniform bound on(C^k)is obtained thanks to Lemma 3.2, settingT^∗=T in the first case (arbitraryC⁰) and setting T^∗= +∞in the second case (C⁰small enough andγ ∈(−d+1,0]). In both cases, for anyT ∈(0, T^∗)we may define the parameterτ by (3.5).

Proof of Lemma 3.2. – Noticing that if, for someC^∗>0, we have

C^jC^∗ for anyj=0, . . . , k, (3.6) thenC^j+1(1+(τ +C^∗<((j+1)t))t)C^j for anyj =0, . . . , k. We deduce that

C^k+1 k

j=1

1+(τ+C^∗<(j+1)t)t

C⁰

exp

kt

0

τ +C^∗<(s)ds

C⁰

exp

τ T +C^∗ T

0

<(s) ds

C⁰, ifkt T . (3.7) Consider first the general case and take C⁰>0 arbitrary. Let choose C^∗ >4C⁰ and T >0 (small enough) so that

exp

C^∗ T

0

<(s) ds

C^∗

2C0

.

Then for τ =ln 2/T and kk^∗ we deduce from (3.7) that C^k+1 C^∗. Thus, by induction, (3.4) holds.

Now, consider the caseγ ∈(−d+1,0](so that<∈L¹(R₊)). We remark that if

C^∗=1, C⁰C¯ :=exp

− ∞

0

<(s) ds

,

and since, for an appropriate choice ofK2(small enough), τ 1

T ∞

T

<(s) ds,

then we also haveC^k+1C^∗and we conclude again by induction. ✷

LEMMA 3.4. – There isK3=K3(α)such that for anyθ ∈ ]0,1[the condition

x(Rx+T Rv)K3θ (3.8)

implies

P ξ^{−k #}−ξ^{−k #}θ ξ^{−k #} ∀x∈BRx, v∈BRv, kk^∗, (3.9) and

PLRvξ^{j #k #}(1+θ )LRvξ^{j #k #} ∀x∈BRx, v∈BRv, k, j+kk^∗. (3.10) Proof. – Letx∈aand write

P ξ^{−k #}−ξ^{−k #}(x, v)

=ξ^{−k #}

|a|

_a−x

expα|x−ktv|²−α|z+(x−ktv)|²−1dz.

Since a−x⊂ [−x, x]^d by definition ofa and x, one has onBRx ×BRv, taking x1,

expα|x−ktv|²−α|z+(x−ktv)|²−1 exp4α(Rx+T Rv)x

−1

4α(Rx+T Rv)xexp4α(Rx+T Rv)x

. The inequality (3.9) follows taking for instanceK₃⁻¹:=4αe^4α.

In order to abreviate the notation we putφ:=(LRvξ^{j #})^{k #}. Letx∈aand write (P φ−φ)(x, v)

= 1

|a|

R^d

a−x

ξ(x+t(j w+kv), w)K(x, z, v, w)ARv(v−w) dw dz,

where, similarly,

K(x, z, v, w)=expαx+t(j v+kw)²−αz+x+t(j v+kw)²−1 4α(Rx+2T Rv)xexp4α(Rx+2T Rv)x

. Hence

(P φ−φ)(x, v)θ

R^d

ξx+t(j w+kv), wARv(v−w) dw=θ φ(x, v). ✷

PROPOSITION 3.5. – For any C⁰ > 0 there exists T^∗ > 0 (defined in Remark 3.3), K4>0 such that, for any choice of discretisation parameters t, x, Rx, Rv, T satisfying

T < T^∗, T RvRx/4, tK4, (3.11) x(Rx+T Rv)T^γ+dK4t, (3.12) the sequence (M^k)k∈N defined by

M^k(x, v)=C^kξ^{−k #}, k=0, . . . , k^∗=E(T /t), (3.13) with(C^k)k∈Ngiven by Lemma 3.2, satisfies

M^k+1P M^k−#+tQ⁺P M^{k #}, P M^k−# on{1, . . . , k_∗} ×B3Rx/4×BRv. (3.14) Moreover, if

0finM⁰ (3.15)

then

0f^kM^k on{1, . . . , k_∗} ×BRx/2×BRv. (3.16) Proof. – Fixk, x, vso thatkt T,x∈B3Rx/4and v∈BRv. Since|x+tv|Rx, one has, according to lemma 3.4, (1.3), (1.4) and the fact thatQ⁺ is a positive operator

Q⁺P M^k−#, P M^k−#=C^k2Q⁺Pξ^{−k #}, Pξ^{−k #} (1+θ )²C^k2Q⁺ξ^{−(k+1) #}, ξ^{−(k+1) #} (1+θ )²C^k2Lξ^−(k+1)#ξ^−(k+1)#.

Since ifK4is chosen small enough thenθ=τ t1, we infer from (3.2) and (3.3) that P M^k−#+tQ⁺P M^k−#, P M^k−#

(1+τ t)C^k+t

C^k2<(k+1)t)ξ^−(k+1)#

C^k+1ξ^−(k+1)#=M^k+1, and (3.14) holds. Let us now assert that

0f^kM^k onB3Rx/4−ktRv×BRv. (3.17) According to (3.15) it is obviously true fork=0. Assume it is true for somek. Then f^k#M^k#onB3Rx/4−(k+1)tRv×BRv and therefore by definitions off^k+1andM^k+1we also have

f^k+1Pf^k−#+tQ⁺Pf^k−#, Pf^k−#

PM^k−#+tQ⁺PM^k−#, PM^k−#M^k+1 onB3Rx/4−(k+1)tRv×BRv. Moreover, we have from (3.15)